A completely different approach (from my other answer) is to use the concept of blossoming, which is explained in these notes written by Lyle Ramshaw. Ramshaw clarified and popularized the blossoming idea, though it was originally developed by de Casteljau in the 1960s.
If $F$ is the blossom of the Bézier curve, then, using the notation from my other answer, we have
$$
P_0 = F(0,0,0) \; ; \; P_1 = F(1,0,0) \; ; \; P_2 = F(1,1,0) \; ; \; P_3 = F(1,1,1)
$$
and
$$
Q_0 = F(0,0,0) \; ; \; Q_1 = F(\tfrac13, \tfrac13, \tfrac13) \; ; \; Q_2 = F(\tfrac23, \tfrac23, \tfrac23) \; ; \; Q_3 = F(1,1,1)
$$
The function $F$ is multi-affine (affine in each variable). The de Casteljau algorithm uses the multi-affine property to calculate $Q_0,Q_1,Q_2,Q_3$ from $P_0,P_1,P_2,P_3$. But it seems that knowing the value of $F$ at any four "general" places will let you calculate its value at any other place, again using the multi-affine property. So, in particular, you can calculate $P_0,P_1,P_2,P_3$ if you know the curve points corresponding to any four parameter values.
Actually, on page 11 of the notes I cited above, we learn that, provided the cubic curve is twisted in space (i.e. not planar), the blossom point $F(u, v,w)$ can be constructed geometrically as the intersection of the curve's osculating planes at the
three points $C(u) =F(u,u,u)$, $C(v) = F(v,v,v)$, and $C(w) = F(w,w,w)$.
Of course, this requires that $u,v,w$ are distinct, so it won't let us calculate the control points $P_1 = F(0,0,1)$ and $P_2 = F(0,1,1)$ directly. But there's a modified rule that works, as explained on page 12 of the notes: the blossom point $F(u, v,v)$ can be constructed geometrically as the intersection of the curve's osculating plane at the
point $C(u) =F(u,u,u)$, and its tangent line at $C(v) = F(v,v,v)$. So, in particular, $P_1 = F(0,0,1)$ is the intersection of tangent line at $C(0)$ and the osculating plane at $C(1)$. Similarly, $P_2 = F(0,1,1)$ is the intersection of tangent line at $C(1)$ and the osculating plane at $C(0)$.